| L(s) = 1 | + 0.208·2-s + 1.89·3-s − 1.95·4-s + 0.394·6-s + 1.28·7-s − 0.824·8-s + 0.588·9-s − 3.70·12-s − 5.45·13-s + 0.268·14-s + 3.74·16-s + 3.93·17-s + 0.122·18-s − 2.24·19-s + 2.44·21-s − 0.711·23-s − 1.56·24-s − 1.13·26-s − 4.56·27-s − 2.52·28-s − 3.65·29-s − 6.46·31-s + 2.42·32-s + 0.818·34-s − 1.15·36-s + 8.26·37-s − 0.467·38-s + ⋯ |
| L(s) = 1 | + 0.147·2-s + 1.09·3-s − 0.978·4-s + 0.161·6-s + 0.487·7-s − 0.291·8-s + 0.196·9-s − 1.06·12-s − 1.51·13-s + 0.0717·14-s + 0.935·16-s + 0.953·17-s + 0.0288·18-s − 0.515·19-s + 0.532·21-s − 0.148·23-s − 0.318·24-s − 0.222·26-s − 0.879·27-s − 0.476·28-s − 0.679·29-s − 1.16·31-s + 0.429·32-s + 0.140·34-s − 0.191·36-s + 1.35·37-s − 0.0758·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.208T + 2T^{2} \) |
| 3 | \( 1 - 1.89T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 0.711T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 - 8.26T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 - 1.31T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 + 9.71T + 53T^{2} \) |
| 59 | \( 1 - 8.91T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 + 4.30T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 6.28T + 89T^{2} \) |
| 97 | \( 1 + 0.303T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.352530650502169149886411867220, −7.80230763770557905337921869407, −7.16998040983689370038687341581, −5.81854300173495979221200797225, −5.15393617228347325411487910387, −4.35891592659975602941896626942, −3.53653337215498187861838665780, −2.72628678643951642411950580702, −1.68770170720250048718109144151, 0,
1.68770170720250048718109144151, 2.72628678643951642411950580702, 3.53653337215498187861838665780, 4.35891592659975602941896626942, 5.15393617228347325411487910387, 5.81854300173495979221200797225, 7.16998040983689370038687341581, 7.80230763770557905337921869407, 8.352530650502169149886411867220
